Hippasus, the Pythagoreans and the Broad Beans (part 2)
In case you missed it, we advise you to read the first part of this article.
The Broad Beans
In the previous episode, we deliberately left out the first commandment. As can be seen from the emphasis and position in the list, the Pythagoreans had, in fact, some difficulties with the broad beans.
In fact, the broad beans are in Pythagoras exactly like kryptonite is in Superman.
Don’t believe it? Then perhaps you do not know that the Pythagoreans believed that the same vital breath that made men alive resided in the broad beans. It was therefore essential not to consume them, also because after eating them this breath would manifest itself freeing itself in the digestive system and exiting the back in a decidedly more mephitic form.
It is no coincidence that the broad beans were made connected to the world of the dead.
Favism is a genetic disease linked to the X chromosome, one of the most common in the world. It results in a deficiency fo an enzyme known as G6PD (which is not the name of a Star Wars droid but stands for glucose-6-phosphate dehydrogenase). Without this enzyme, red blood cells are unable to resist oxidation and are destroyed.
There can be devastating consequences for the individual such as the complete destruction of red blood cells (acute hemolysis) and jaundice. In short, you risk becoming like a Simpsons character.
Just as the First Commandment says, those with favism should not eat beans, to begin with. In addition to this, commonly used substances such as naphthalene (so the cabinets become a nice feast for moths), verbena and drugs such as analgesics, antipyretics, antimalarials, some chemotherapy and antibiotics that can aggravate the cannot be taken or inhaled condition by increasing the oxidation processes in the body.
People affected by this disease who want to end their lives can also devote themselves to taking mega doses of vitamin C.
It seems that in the Crotone area this disease was particularly widespread in the times of Pythagoras, and therefore the Pythagoreans simply tried to protect their followers from this pathology whose nature they ignored. Be that as it may, this prohibition was of crucial importance, so much so that numerous Pythagoreans and perhaps even Pythagoras himself had died.
The legends tell of how the Pythagoreans had the vice of supporting several aristocratic governments.
Sometimes, however, it happened that these governments were overthrown by the democrats, who promptly hunt down the followers of the triangles with torches and pitchforks.
It so happened that Pythagoras, having briskly fled to escape capture, found himself facing an insurmountable field of broad beans. It seems that rather than cross it and thus come into contact with the nasty pods, he preferred to be joined by his pursuers.
There is even an anecdote that tells us how once Pythagoras even managed to convince a bull to refrain from eating broad beans. But his oratory skills have been less effective in the last juncture of his life. Upon reaching him, his fellow citizens barbarously slaughtered him.
There is also news of a couple of disciples of Pythagoras, Milliade of Crotone and his wife who, fleeing the angry crowd after the fall of the aristocratic government, they were finally reached by being in front of a field of broad beans.
The attackers, having a surge of compassion for the pregnant woman, decided not to smear the pavement with their blood but to lead them rather in the presence of the new tyrant Dionysius. He offered them life-saving if only they revealed the reason that pushed them not to cross the field. But the couple wisely chose to die.
Hippasus
But after this beautiful narration on the bizarre adventures of the Pythagoreans, we come to the point of the question: what happened to them? Let’s say that in addition to the tragic events involving the beans, the Pythagoreans had to face a far more serious crisis, a crisis from which they could not recover. What can be worse than broad beans? — only the radicals, which are not other plants, now let’s talk about numbers!
It seems that it all began when our Lucanian (means, originally from Basilicata, an Italian region) hero found himself studying the diagonal of a square. And what’s wrong? — you may foolishly think.
Nothing, except that dear Hippasus accidentally discovered something that can be expressed in these clear and simple terms:
the side and the diagonal of the square are immeasurable magnitudes.
In practice, if you try to divide the diagonal by the side of a square, a nice number will come up:
Try to follow us, it’s not so difficult. Suppose you have a square with side l, the diagonal d can be calculated with the Pythagorean theorem.
Or
And therefore, from the ratio between the diagonal and the side we get
At this point, it must be said that the Pythagoreans had somewhat particular ideas about numbers. They were only happy with those numbers that we call natural today and the ratios that can be obtained from these ones (rational numbers).
The natural numbers are 1, 2, 3, … while the ratios that we can obtain with these numbers are for example 1/2, 5/6, 7/4 which today we also express in decimal form 0.5, 0.83333, 1.75. For the Pythagoreans there were no negative numbers (like -21 or -0.3242), much less there could be something that went beyond the fractions.
Yes, because the square root of 2 (and many other things that we now consider numbers) cannot be represented as the ratio between two integers. And we can prove it! But we leave these somewhat complex things for another time.
The fact that no other numbers could exist outside of the natural ones and their ratio was taken for granted and had to be carved in the heads of the adepts just as the maxim
EVERYTHING IS NUMBER.
The latter makes it clear how deeply the Pythagoreans were convinced that the whole universe could be represented by numbers.
It is not much different from what modern physicists think. Except that they are not discouraged by the unexpected and if something does not work, it often means that this allows us to expand our knowledge about a phenomenon.
The Pythagoreans, however, must not have taken the discovery of the square root of 2 in this light and it also seems that the dear Hippasus went to spread the result of his studies around. And if the existence of the square root of 2 should not be liked at all, imagine how well the gesture of Hippasus must have taken. Spreading news that undermines all the knowledge gathered by Pythagoras and his followers.
Some say that the followers of Pythagoras erected a tombstone and treated Hippasus as if he were dead. Others instead claim that somehow Hippasus went to lecture the Ionian fishes (either because they threw it there or because someone made the boat he was travelling sink on).
And here we are! The Hippasus’s Fishes come into play, always ready to hear what our unfortunate mathematician tells us and happy to be able to reciprocate by sharing old and new discoveries from the scientific world.
Sources
Carl B. Boyer, Uta C. Merzbach, (2010), A History of Mathematics (3rd ed.), Wiley, ISBN 978–0–470–52548–7
Roger L. Cooke, (2013), The history of mathematics: a brief course (3rd ed.), Wiley, ISBN 978–1–118–21756–6
